If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$

Question

If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$

Attempt: $|H|=p^k \implies |G|=p^{n_1} q^{n_2} r^{n_3} \cdots~~|~~n_1 \geq k$

Now, $H$ is also a normal sub group of $G$ and we need to show that $H$ is contained in every $p$ sylow subgroup of $G$ which means in all of $H_{p^{n_1}}, H_{q^{n_2}},H_{r^{n_3}}, \cdots $

Unfortunately, I am not able to decide the strategy to move ahead. How do I move forward?

Thank you for your help.

Answer 1

Hints:

1. Since $H$ is normal, what do you know about the conjugates of $H$?

2. What do the Sylow theorems tell you about how different Sylow $p$-subgroups are related, for any given prime $p$?

Answer 2

Another strategy: if $G$ is a finite group and $N \lhd G,$ then whenever $X$ is a subgroup of $G,$ then $XN = NX$ is a subgroup of $G$ of order $\frac{|X||N|}{|X \cap N|}.$ Now consider the case that $H = N$ and $X \in {\rm Syl}_{p}(G).$